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Curvature oscillations and linear electro-optical effect in a surface layer of a nematic liquid crystal
L. Blinov, Geoffroy Durand, S. Yablonsky
To cite this version:
L. Blinov, Geoffroy Durand, S. Yablonsky. Curvature oscillations and linear electro-optical effect in a surface layer of a nematic liquid crystal. Journal de Physique II, EDP Sciences, 1992, 2 (5), pp.1287-1297. �10.1051/jp2:1992200�. �jpa-00247726�
Classification Physics Abstracts
61.30 68,10 68.45
Curvature oscillations and linear electro-optical ewect in a surface layer of a nematic liquid crystal
L. M. Blinov (I. *), G. Durand (I) and S. V. Yablonsky (2)
(1) Laboratoire de Physique des Solides, Bit. 510, Universit6 Paris-Sud, 91405 Orsay, France (~) Institute of Crystallography, U-S-S-R Academy of Sciences, 59 Leninsky prospect, 117333
Moscow, Russia
(Received 19 April 1991, revised 27 November 1991, accepted 23 December 1991)
Rdsumk.-Des oscillations de courbure amorties sont excitdes h l'interface entre un cristal
liquide ndmatique et un substrat solide, par le couplage lindaire entre un champ dlectrique
altematif et la polarisation flexodlectrique du ndmatique. Les ondes sont ddtectdes par une
technique ellipsomdtrique. On observe une oscillation du directeur h la fr6quence du champ.
L'amplitude et le temps caractdristique de cette oscillation sont ddfinis par les propridtds visco£lastiques et flexo£lectriques du milieu, et le recouvrement de l'onde de courbure avec le
profil de l'onde optique £vanescente de mesure. Un modble simple est discut6, en bon accord
avec les expdriences, faites sur
un m61ange compensd de MBBA (d'anisotropie didlectrique nulle)
et sur du SCB. L'£nergie d'ancrage sur l'dlectrode dvaporde SiO oblique, et la somme ej + ei des coefficients flexo£lectriques sont mesur£es, indiquant une baisse du parambtre d'ordre
en surface.
Abstract. Damped curvature oscillations are excited at the surface between a nematic liquid crystal and a solid substrate, from the linear coupling of an AC electric field with the flexoelectric
properties of the medium. The waves are detected by a modulation ellipsometry technique.
Linear-in-field oscillations of the director in the surface nematic layer have been observed. The
amplitude and characteristic response time of the oscillations are defined by visco-elastic and flexoelectric properties of the medium and the overlaping of the curvature wave with the profile
of the sampling optical evanescent wave. A simple model is discussed which is in good agreement with experiments performed on a compensated MBBA mixture (with zero dielectric anisotropy)
and SCB. The anchoring energy for the nematic in contact with obliquely evaporated SiO layer, as
well as the sum ej + ei of the flexoelectric coefficients are measured, indicating a surface order parameter smaller than its bulk value.
(*) Present address : Institute of Crystallography, Academy of Sciences, 59 Leninsky prospect, 117333 Moscow, Russia.
Introduction.
Nematic liquid crystals are centrosymmetric media. However, when they are distorted, this symmetry is broken. A flexoelectric polarization P occurs which is a linear function of the
curvature [I].
P= eindivn+e~(curln)x n (I)
where n is the director (n~
= I) and ei,~ are flexoelectric coefficients. When an external electric field is applied to a liquid crystal cell, it can interact with the curvature-induced
polarization through the corresponding flexoelectric energy W~ = E P. It is well known, however, that in the one-dimensional case, when the field (uniform) and the distortion are
constrained to be in the same (say x, z) plane, the whole flexoelectric torque is integrated out of the volume resulting in a surface flexoelectric torque only [2]. Thus, if one applies an
electric field to a cell with symmetric boundary conditions, at equilibrium, two flexoelectric torques from the opposite boundaries will cancel each other and no low field distortion can be
observed. High field instabilities have been predicted [3] and observed [4-6]. For a more
general three dimensional case the situation is different. The anisotropic flexoelectric torque (ei e~) gives rise to bulk effects [7].
If, however, one investigates the curvature (flexo-) electricity in the dynamical regime, opposite symmetrical boundaries can be decoupled. Of course, it depends on the characteristic
wavelength of the distortion which must be shorter than the cell thickness. For example, if an
a.c. electric field is applied to a bulk nematic, it must create an oscillating flexoelectric torque
at the surface and a surface induced curvature oscillation which will decay toward the bulk.
For cells thick enough there will be no interference with the corresponding curvature waves
going in opposite directions from the opposite boundaries.
The aim of the present paper is to apply the modulation ellipsometry technique [8] to the
investigation of the field induced curvature oscillations at the interface between a solid substrate and a nematic liquid crystal. In this case, the linear coupling of the extemal field with the flexoelectric properties of the nematic liquid crystal results in a linear-in-field
director oscillation at the fundamental (not double) frequency of the applied field. For high enough frequencies (in fact higher than a few Hz), the curvature wavelength is short enough.
In practice, one can neglect the influence of the opposite surface, even for cells as thin as 10 ~Lm. The technique, in principle, allows one to measure the anchoring energy of a nematic at the interface with a solid substrate, to investigate surface ordering, screening of the
flexoelectricity by ionic charges, etc.
Theory.
Let us discuss in more details the simplest model for the curvature oscillations electrically
induced at the interface between a nematic and a solid substrate. Figure shows the geometry discussed. We consider a semi-infinite pure dielectric nematic medium (no ionic processes)
with negligible dielectric anisotropy e~
= 0. The distortion is constrained to be only in the X, Z plane. The torque balance equation for the bulk contains only the elastic and viscous terms
and has the familiar form of a diffusion equation
-K~~+~((=o (2)
where 0 = 00 + 30 includes a static pretilt angle 00 and the dynamic variation Be induced from the surface Assume for simplicity 00 uniform in the cell. K is a combination of the bend
So
~~~~
X
~ i
) fl
j 0
0/
~
(
~ b
0 ~
x/(2q~)
Fig, I. Maximum profile of the curvature wave 8H in the nematic. The Slid ~urface i; at
z 0 a) evanescent optical wave for q~ A « I b) evanescent optical wave for q~ A i. Insert :
definition of angles.
and splay elastic moduli, and
7~ is an effective viscosity (mostly rotational viscosity, with some correction from back-flow). Taking the Fourier transform for 30
30
=
30(z)exp(iwt)
= 30~exp(iwt) exp(iqz)
we obtain the dispersion relationship :
Kq~= iw7~
=
0 (3)
with solution
q" 1-1+ij(] ~~~= j-1+ijq~. (4)
Equation (2) describes the decaying curvature wave with the space variation :
Be (z) = Be ~exp (- q~ z) exp (- iq~ z) (5)
The surface amplitude of the director oscillation Be ~ is still to be calculated. 30~ will be
generated at the surface by the flexoelectric effect and plays the role of a boundary condition for equation (2).
To calculate the magnitude 30~, we write the torque balance equation at the surface. It includes a destabilizing, linear-in-field flexoelectric term 1I~~~~ and two stabilizing ones, one
coming from the anchoring of the director at the surface and the other taking into account the visco-elastic reaction of the medium (integrated out from the bulk to the surface). For the director having the form (sin 0, 0, cos 0), equation (I) gives :
~~~~° t ~ ~~~ 2° ~ (6)
where 0~
= 00 + Be ~. We use a sufficient low field E
= E~ exp (iwt), and an oblique initial director orientation (00 # 0, 00 # w/2) so that we remain in the low distortion approximation
Be ~< Ho. In equation (6), we can just replace 0~ by 00. This approximation is not valid for oo = 0 (homeotropic orientation), or oo = w/2 (planar orientation). In this case, 1I~~~~ would
write as :
"flexo " ~~ i ~~E Sin 23°~ (6~)
we discuss this situation later.
For small surface angular deviations around the equilibrium orientation 00, the stabilizing torque coming from the anchoring energy may be taken in Ihe Rapini form :
H~~~~ =
~ ~
(3 0~)~ = ~ Be ~ (7)
b(30~) 2 L L
where L
= K/Ws is a characterisIic length defining the anchoring energy W~. The viscoelasIic surface reaction of the liquid crystal can be integrated from equation (2). Using equation (4),
we find :
H~_~j=-K ~~~=-Kiq3H~= (1+I)q~K3H~ (8)
The torque balance equation for the amplitude of the director oscillation at the surface reads :
In the previously described approximaIion, the amplitude of Ihe direcIor oscillaIions at the surface is :
~
(ej + e~) E~ sin 2 Ho
3
= (10)
2 KIL~ + (i + I) q~j
In the case of the strong anchoring, q~ « L
~, equation (10) predicts surface oscillations in
phase with the electric field, with amplitude :
3 ~
= (ej + e~) E~ sin 2 Ho/2 W~ I I)
A decrease in the anchoring energy as well as an increase in frequency w result in a
crossover of the two characteristics lengths L and qj and in a phase shift between the field
and the director oscillation at the surface. In the limit of a very weak anchoring, the
oscillations should present a phase lag of 45° with the field, with amplitude :
~
(ej + e~) E~ sin 2 Ho
~~ 2(/) Kq~ ~~~~
As soon as q~ is known, one can calculate the anchoring energy from the amplitude
3H~ (for known ej ~) or from the phase shift predicted by equation (10). Note that
(et + e~)/2
=
Ssi, where Ss is the surface order parameter modulus, and I the material
flexoelectric constant. Equations (5) and (10) define the bulk curvature oscillation in a nematic liquid crystal induced by an electric field from the surface. This oscillation can be detected optically in the form of a linear-in-field electro-optical effect. The best way to probe
a surface layer optically is to use the light total reflexion technique. In the case of a light
incidence at a grazing angle (higher than total reflexion angles for both polarizations), an evanescent wave of amplitude
-w
exp(- q~ z) is created close to the solid-nematic interface.
Note that one can chose such a geometry that only the extraordinary wave is sensitive to the
expected change in nematic director tilt. Calling n~(~ l.72) the corresponding index of refraction, at grazing incidence, q~ is approximately given by :
q~ =
~ "
(n~ n/)~'~
A
where n is the (large) index of the glass, and A the vacuum wavelength of light. The
polarization of the evanescent wave is almost normal to the surface. In principle, it is possible
to calculate the director deviation angle from the changes in the azimuthal angle of the totally
reflected light polarization ellipse [8].
The ellipsometric signal produced by the director tilt is proportional to the convolution of the static shape of the evanescent wave and the shape of the curvature wave (Fig. I). The
resulting integral :
m
I
= o exp [- (q~ + q~ ) z cos q~ z dz
normalized to its maximum value for q~ = 0, (I.e. Io
=
I/q~) is easely calculated as : 1 q>(qw + q>)
IO q$ + (qw + q> )~
1/Io is a decreasing function of q~, from I (q~
=
0) to zero (q~-co). One finds
I/Io
=
1/2 for q~ = q~/ / l/A. For low frequency excitation, when q~ « A , the optical ellipsometry probes the actual amplitude 3H~ at the surface. In the high frequency limit when q~ » A the director oscillations and the corresponding changes in the refractive indices are averaged out and we expect a dramatic decrease in the amplitude of the linear electro-optical
effect. Since q~ is constant and q~ (for a given material) depends only on the frequency of the
applied field, the previously discussed frequency crossover will determine the characteristic time of the linear electro-optical effect. Taking as a qualitative criterion I/Io = 1/2, we arrive at :
f~
= w~/2 w = K/(w7~A~). (13)
Note that no dephasing is expected from this geometrical convolution.
Let us discuss now briefly the homeotropic alignment situation Ho = 0 (or equivalently the
planar one 00
= w/2). The surface torque equation is :
For small field, the solution is 3 0~
= 0. There exist a field threshold :
~~
K[L~~+(I+I)q~]
E~= (14)
~l+~3
above which a non zero oscillating Be
~, of arbitrary sign, is allowed. This would be in fact the AC version of the DC flexoelectric instability long ago predicted by Helfrich [3], and more
recently observed [4-6]. We shall not discuss this situation any further, since it does not
correspond to our experimental geometry.
The very simple model presented above can be generalized to take into account various other phenomena :
I) For very low frequencies and very thin cells, the destructive interference of the waves
originating from the two opposite surfaces must be taken into account. This will result in the
vanishing of all unidimensional distortions under discussion.
2) The quadratic coupling of the field with dielectric anisotropy must be taken into account
for the case e~ #0. It will result in a non-linear equation for the curvature and more
complicated shape of the curvatures waves.
3) Ionic processes can screen the electric field in the sample, and increase the field near the surface. This effect must be taken into account for rather conductive substances or when
qj compares with the Debye screening length.
4) Non-symmetric boundary conditions gives rise to a d.c. component of the distortion which is not averaged out and would dominate at low frequencies.
Experimental technique.
A scheme of our experimental set-up is shown in figure 2. The principal element is a liquid crystal cell consisting of a prism with high refractive index (n =1.806) and an optically polished glass plate, both covered with SnO~ electrodes. A nematic liquid crystal of zero
dielectric anisotropy e~ at room temperature is prepared, by mixing MBBA (methoxy
benzylidene butyl aniline e~
= 0.5) with 4-heptyl-4'-cyanophenyl benzoate e~ = + 9). The mixture has a measured e~ = + 0.07. This nematic is oriented homogeneously (by rubbing a polyvinyl alcohol layer on the electrodes), homeotropically (with chromium distearylchloride
surfactant) or with a molecular tilt 00
=
72° (using SiO evaporation at a grazing angle). The cell is installed on an optical bench and an ellipsometry technique is used to detect angular
oscillations of the nematic orientation at the surface.
8
5 3
~ / 6
Fig. 2. The experimental set-up. I laser, 2 : polaroids, 3 liquid crystal cell, 4 : AM plates, 5 : photodiode, 6 lock-in amplifier, 7 : oscilloscope, 8 function generator.
The principle of'the opIical modulaIion elhpsomeIry meIhod we use has been described m details in a previous paper [8]. Beyond total intemal reflexion (TIR) a light beam totally
reflected from isotropic media undergoes a phase shift which depends on its polarization,
transverse electric (TE) or transverse magnetic (TM), and on the refractive indices of the
materials. Using an interference technique, one can measure the phase difference induced on
the two beams after TIR. One uses for instance an incident laser beam with a linear
polarization rotated by 45° from the place of incidence, to produce two (TE and TM) beams
of equal amplitudes and phases. After TIR, the phase diit'erence produce~ an elliptically polarized reflected beam. The measurement of the ellipticity allows the determination of the
refractive index (or the thickness) of the material at contact with the high index glass prism.
To apply this standard ellipsometric technique to the nematic liquid crystal, which is optically anisotropic, we orient the director (planar or oblique) in the plane of incidence, which is now a symmetry plane for the whole system. (The director orientation is known from the rubbing or the evaporation directions). The two TE and TM beams become respectively ordinary (o) and extraordinary (e) polarized. If one knows the corresponding main refractive
indices n~ and n~, the measured phase shift difference from the two TIR beams allows the calculation of the surface director tilt for a thick enough and uniformly oriented nematic interfacial layer. When an AC electric field is applied on the nematic material, one expects a
forced oscillation of the surface director, I.e. a modulation of the TIR induced ellipticity, synchronous with the field. The surface director oscillation amplitude is calculated from the
measured ellipticity modulation.
In practice (Fig. 2), we illuminate the glass-nematic interface with a He-Ne laser beam
(A = 6 328 h). The angle of light incidence (#
= 80°) is larger than the total reflection
angles for both (o) and (e) laser polarizations. This provides a relatively short penetration length for the evanescent waves. A quarter wave plate converts the linear laser polarization
into a circular one. A polarizer is then oriented at 45° from the incidence plane, to generate the two equal amplitude (o) and (e) beams. The totally reflected laser beam is analyzed by a
quarter wave plate and a rotating analyzer. The AM wave has one of its eigen axes rotated at 45° from the plane of incidence, to convert the elliptical polarization into a linear one. The rotation of the analyzer from this 45° orientation, which results in maximum transmitted
intensity, measures the phase shift between the two reflected beams, I.e. would allow in
principle to control the average surface tilt angle 00. The surface orientation is modulated by
an electric field E
= E~ cos wt from a signal generator, with frequency f
=
w/2 w from 0.5 Hz to 200 Hz, applied to the cell electrodes. The linear in field electro-optical modulation of the reflected beam appears at the frequency f, the first harmonic of the field. To observe this modulation, the reflected beam intensity is detected by a photodiode and recorded on a
lock-in amplifier, synchronously with E.
To evaluate the angle of deviation of the surface director, we measure first the DC intensity
of the reflected light versus the angular orientation of the analyzer (Fig. 3, curve I), with respect to the plane of incidence. In absence of reflection induced phase change, the Malus
law would give a sine wave with maximum and minimum at 45° and 135°. Here, we observe a
phase shift of 20°. Using the formula of reference [8], we can check the value 00
=
72°, in agreement with the expected value.
We now measure the amplitude of the AC detected current at frequency f, versus the
analyzer angular orientation (Fig. 3, curve 2). This amplitude is proportional to the derivative
of curve I, I.e. its maximum is dephased by 45°. Knowing the slope of curve I, we can
determine the absolute change in phase induced by the electric field [8]. We then calculate the
corresponding change in refractive index for the extraordinary wave and finally the director deviation Be ~ close to the prism surface. All our reported data have been taken by keeping
the orientation of the analyzer along the direction of maximum AC signal.
The linear-in-field electro-optical response was also observed for pure MBBA e~ =
0.5 and SCB (pentyl cyano biphenyl) (e~ = + 8). In the case of small applied field the behaviour of the latter is very similar to that of zero dielectric anisotropy MBBA. For higher field the
quadratic electro-optical effect which, in all cases, is observed at the double frequency of the
applied field, influenced the field response at the first harmonic. We have also performed comparative pulse measurements of the linear effect and the Freedericks transition (for dopped MBBA and SCB) which allow the absolute sign of the linear effect to be determined.
